Thermodynamic Temperatures of High-Temperature Fixed Points: Uncertainties Due to Temperature Drop and Emissivity

This study forms part of the European Metrology Research Programme project implementing the New Kelvin to assign thermodynamic temperatures to a selected set of high-temperature fixed points (HTFPs), Cu, Co–C, Pt–C, and Re–C. A realistic thermal model of these HTFPs, developed in finite volume software ANSYS FLUENT, was constructed to quantify the uncertainty associated with the temperature drop across the back wall of the cell. In addition, the widely applied software package, STEEP3 was used to investigate the influence of cell emissivity. The temperature drop, ΔT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta T$$\end{document}, relates to the temperature difference due to the net loss of heat from the aperture of the cavity between the back wall of the cavity, viewed by the thermometer, defining the radiance temperature, and the solid–liquid interface of the alloy, defining the transition temperature of the HTFP. The actual value of ΔT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta T$$\end{document} can be used either as a correction (with associated uncertainty) to thermodynamic temperature evaluations of HTFPs, or as an uncertainty contribution to the overall estimated uncertainty. In addition, the effect of a range of furnace temperature profiles on the temperature drop was calculated and found to be negligible for Cu, Co–C, and Pt–C and small only for Re–C. The effective isothermal emissivity (εeff)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon _{\mathrm{eff}})$$\end{document} is calculated over the wavelength range from 450 nm to 850 nm for different assumed values of surface emissivity. Even when furnace temperature profiles are taken into account, the estimated emissivities change only slightly from the effective isothermal emissivity of the bare cell. These emissivity calculations are used to estimate the uncertainty in the temperature assignment due to the uncertainty in the emissivity of the blackbody.


Introduction
This study forms part of the European Metrology Research Programme project Implementing the New Kelvin [1,2]. This work inputs into workpackage one (WP1) of that project to assign thermodynamic temperatures to a selected set of high-temperature fixed points (HTFPs) [3], namely, the eutectics Co-C (1597 K), Pt-C (2011 K), and Re-C (2747 K). In addition, the thermodynamic temperature of the Cu point (1358 K) will be measured. The HTFP cells selected for the temperature assignment will be measured by nine different institutes and the temperature values assembled into one consensus temperature as detailed in [4]. When agreeing on a consensus value, it is important to minimize uncertainties, to understand where potential systematic differences between national metrology institutes (NMIs) come from and to identify the differences that are intrinsic to HTFP cells.
Since the definition of the uncertainty budgets for realization of scales by radiation thermometry [5], several papers refining the estimates of the temperature drop and emissivity uncertainties have been published [6][7][8]. The aim of this paper was to enable definitive assessment of the uncertainty due to these two effects, as contributors to the HTFP temperature assignment uncertainty.
In this work, a realistic thermal model of the actual HTFPs used was constructed to quantify the uncertainty associated with the temperature drop, T , across the back wall of the cavity in the HTFP cell. In addition, specialist software, STEEP3 [9], was used to investigate the effect of the emissivity on thermodynamic temperature determination. The estimated corrections for the emissivity and temperature drop are small compared to other sources of uncertainty in the thermodynamic temperature assignment, and because of that, they are likely to be included in the overall uncertainty assessment as Type B uncertainties rather than corrections [4].

Modeling HTFP Cells and Furnace
The HTFP cells were provided by six national measurement institutes as detailed in Table 1, which gives their dimensions as well. The cells of Physikalisch-Technische Bundesanstalt, PTB (Germany); National Metrology Institute of Japan, NMIJ (Japan); and National Institute of Metrology, NIM (China) have the same hybrid design, implying the ingot being surrounded by a graphite sleeve, separated from the crucible by two layers of grafoil. All-Russian Research Institute of Optical and Physical Measurements, VNIIOFI (Russia) provided three different cell designs, Laboratoire National de Métrologie et d'Essais and Conservatoire National des Arts et Metiers, LNE-CNAM Units in mm (France) used the same (hybrid) design for all their cells, and All-Russian Mendeleyev Metrology Institute, VNIIM (Russia) cell had its own design. All the cells were modeled in the same furnace, Type Nagano VR10-A23, one of the furnaces that will typically be used to realize the HTFPs for the assignment of thermodynamic temperature.
The numerical analysis of the melting process was performed by a 2D axisymmetric finite volume model, developed in ANSYS FLUENT 1 . Temperature profiles considered along the furnace tube were (a) uniform temperature profiles at the temperature T E + (T E ), where T E refers to the eutectic temperature and (T E ) to the offset inducing the melt, which is taken to be from 4 K for Cu up to 20 K for Re-C; and (b) temperature profiles likely to be more extreme than that encountered in actual furnaces were also modeled, the assumption being that the reality will be somewhere between the two extremes modeled.
The thermophysical properties of the materials used in the thermal modeling are detailed in Table 2.

Temperature Drop Calculations
The temperature drop, T , is related to the temperature difference-due to the net loss of heat from the aperture of the cavity between the back wall of the cavity, viewed by the thermometer defining the radiance temperature, and the solid-liquid interface of the alloy, defining the transition temperature of the HTFP. More specifically, in Melting heat (J · kg −1 ) this work T is defined as this difference at the inflection point of the melting curve, so it includes the temperature drop over the still solid part of the ingot, assuming melting from the outside of the ingot only. It is obvious that T increases with T E but the modeling performed here shows that its value is always small compared to other contributions to the measurement uncertainty. The temperature drop, as defined, was calculated for all the cells inside the Nagano furnace with a linear (i.e., uniform) temperature profile along the cell and a sineshaped temperature profile from the cell aperture to the furnace aperture, as specified in Table 3. It is anticipated that these are extreme cases and that the actual furnace temperature profile likely lies between these two. Earlier work [6] has shown that the actual profile itself does not have a significant influence on the resultant estimated temperature drop values. The temperature profile is maintained in shape, but scaled in value for different transition temperatures. As an example, the furnace temperature profile no. 12 is plotted in Fig. 1. The calculated value for this rapid decrease in temperature can be compared with the reference temperature drop for a completely uniform furnace (labeled constant profile in Tables 3, 4, 5).
With all these different furnace configurations, the melting front of the alloy was modeled and the temperature drop and melting plateaus were calculated.

Effective Emissivity Calculations
The effective emissivity was calculated over the wavelength range from 450 nm to 850 nm by STEEP3 v1.3 [9] both for the HTFP cells installed in the furnace, and, for reference, for bare HTFP cells (no furnace and with the crucible side wall held at uniform temperature). The surface emissivity for the graphite was assumed to take the conservative range of values of 0.80 to 0.90.
The study reported in [7] showed there would be only small levels of deviation from the ideal blackbody case given reasonable assumptions concerning the furnace temperature profile.
The internal cavity configuration of the cell was the same for NIM, NMIJ, PTB, and LNE, so they presented the same effective emissivity when modeled. The VNIIM and the three VNIIOFI cells differed in cavity length and aperture, so four different models were constructed so as to calculate the effective emissivity of those cells.
The furnace temperature profile used in STEEP3 was derived as an output of the ANSYS FLUENT numerical modeling using the furnace temperature profiles described in Table 3 from the front of the blackbody cavity. The fixed-point blackbody was assumed to be isothermal which is reasonable during a phase transition.

Temperature Drop Estimates
The temperature drop of the cells with the different furnace temperature profiles is given in Table 4. A consistency check was performed for these results by comparing them with results obtained by previous modeling [8]. The results presented here are obtained from a time-dependent model used to calculate the temperature drop, as defined in Sect. 2.2, at the point of inflection of the melting plateau. These results were found to be very close to the previous ones modeled under steady-state conditions and for similar cells and furnace.
By way of example, Fig. 2 shows the temperature at both sides of the back wall of the cavity during melting with furnace profile no. 13 (uniform temperature profile) for the Re-C NIM cell (11-3). Figure 3 represents the melting plateaus for the same cell (11-3), measured at the back wall of the cavity for the different furnace profiles. Figure 2 shows that the temperature drop is effectively constant over most of the melting curve. Units in kelvin a Furnace temperature profiles described in Table 3 Figure 3 shows that the different furnace profiles do not significantly affect the temperature drop value but do, as expected, influence the melting plateau durations and the position of the points of inflection. It was found that for all the fixed-point cells modeled, the magnitude of the temperature drop was much smaller than other likely sources of uncertainty, for example, at the Re-C point, the best radiometric uncertainty is very likely to be in excess of 0.4 K (k = 2), which is five times larger than the estimate of the temperature drop. For the worst case estimate of 0.096 K (Table 4), this translates to a Type B expanded uncertainty component of u( T ) = 2 · 0.096/ √ 3 = 0.11 K (k = 2), which is much lower than the estimated best radiometric uncertainties.
The uncertainties due to the temperature drop for the fixed points and the different furnace temperature profiles have been calculated, as such, and are presented in Table 5.
The estimated values for the temperature drop are, in relative terms, even smaller compared to the best radiometric uncertainties for the other HTFPs and, hence, the temperature drop values are likely to be included as Type B estimates in the uncertainty budget for the thermodynamic temperature assignment rather than corrections. Units in kelvin a Furnace temperature profiles described in Table 3 It is clear by examining the values in Table 4 that the estimated values of the temperature drop for Cu, Co-C, and Pt-C were essentially immune to the difference in the furnace temperature profile. In the case of Re-C, there was a small influence observed, but this was insignificant compared to the value itself.
From the Re-C temperature drop values some implications can be drawn, referenced in previous studies [6]. The lowest values correspond to the VNIIOFI (no. 17) cell, which has the thinnest back and side walls (1.5 mm) and one of the longest internal cavities (49 mm). Both effects, thinner walls and larger cavities, contribute to decrease the temperature drop on the cavity back wall.

Effective Emissivity Estimates
The emissivity of the blackbody cavity could be affected by two main variables, the variation in the surface emissivity of the graphite used to manufacture the blackbody  cavity and the furnace gradient along and in front of the blackbody cavity. Both these effects were investigated for all the fixed points used in the thermodynamic temperature assignment. Note that in this investigation it was assumed that the cavity itself, during the phase transition, was uniform in temperature because of an intact solid-liquid interface (assuming the melt begins from the outside of the ingot and propagates inwardly) during melting.

Effect of the Isothermal Emissivity of the Fixed Point and the Assumed Uncertainty in the Wall Emissivity
The isothermal emissivity for a range of the blackbody cavities was calculated using STEEP3 for surface emissivity values of 0.8 and 0.9, this being considered the likely range of emissivity values for graphite. Table 6 gives the results of these calculations.
To derive the fixed-point thermodynamic temperature from the radiance temperature, the emissivity of the cavity will have to be corrected for. That correction will be performed as part of the calculation of the single temperature value for each fixed  [4,10]. For illustrative purposes it is possible to use the isothermal emissivity values in Table 6 to estimate the magnitude of the correction and also its uncertainty.
To estimate the uncertainty of the correction consider the emissivity values for the LNE-NIM-PTB design of the blackbody. This was chosen as the most extreme case as having the widest range of isothermal emissivity values. The uncertainty can be considered as a Type B evaluation based on the difference between the emissivity values, i.e., 0.000 21. This translates via T = (λT 2 E /c 2 ) ε, to 0.025 K (k = 1) uncertainty at the Re-C point for a radiance measurement at 800 nm. An estimate of the temperature correction itself can be derived from the departure from blackbody conditions-in this case the difference from 1 (blackbody) for the average emissivity of the two extremes, 0.999 715. This gives a correction of +0.12 K at the Re-C point for an 800 nm radiometer. For all other fixed points the correction for emissivity and its associated uncertainty are less. The conclusion that can be reached is that the correction for emissivity is small compared to other uncertainty sources, but not so small as to be insignificant. For illustrative purposes, Table 7 gives the likely magnitude of the estimated corrections and uncertainties for all the fixed points used for thermodynamic temperature assignment.

Effect of Furnace Gradients in Front of the Blackbody Cavity
Besides the effect of the isothermal emissivity of the actual crucible blackbody cavity described above, there is a possible influence of the furnace on the emissivity of the fixed point. This was investigated through further modeling with STEEP3 including the shape and temperature profile of the furnace in front of the cavity. Two extreme cases were modeled that for a uniform furnace extend from the front of the cavity to a furnace aperture (140 mm away) and that for a furnace with a temperature drop that follows a sine profile decline from the front of the cavity (at the fixed point temperature) to quasi-ambient conditions at the furnace aperture. The aperture diameter of the furnace was 24 mm, and the emissivity of a 1 mm diameter region in the central region of the blackbody back wall was calculated. The assumed surface emissivity for the graphite was 0.85.
It was found that the effect of the furnace on the calculated emissivity values of all the cells was very small-with very little difference from the isothermal case < 0.00 002. This is such a small effect that the uncertainty approach described in the section above essentially covers any uncertainty due to the influence of the furnace on the emissivity.

Conclusions
The most important conclusion is that there is only a weak dependence of the values of the temperature drop and the effective emissivity with the participant cell design and also with the typical temperature profile expected in the furnace. This means that the anticipated uncertainties associated with the temperature drop across the back wall of the HTFP cell and with the effective emissivity of the cavity are significantly lower than the expected radiometric uncertainties [4]. In the light of these findings, it is recommended that no correction be applied for either of these two effects and, instead, because they are small compared to other uncertainty sources, the numbers presented in this paper enter the uncertainty calculation [10] as a Type B evaluation.
The actual value of the temperature drop scales with the nominal temperature of the HTFP; nevertheless, it is lower than 0.100 K at the Re-C point and lower for the other fixed points.
The analytic expression given in [5] over-estimates the temperature drop effect. As a consequence of this work, more realistic values for the temperature drop have been determined, and this will ultimately lead to better estimates of the HTFP temperatures with potential consequences to global temperature metrology.